Using Levy Processes to Model Return Innovations

7/30/2019 Using Levy Processes to Model Return Innovations

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Using Levy Processes to Model Return Innovations

Liuren Wu

Zicklin School of Business, Baruch College

Option Pricing

Liuren Wu (Baruch) Levy Processes Option Pricing 1 / 32
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Outline

1 Levy processes

2 Levy characteristics

3 Examples

4 Evidence

5 Jump design

6 Economic implications



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Outline

1 Levy processes


3 Examples

4 Evidence

5 Jump design


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Levy processes

A Levy process is a continuous-time process that generates stationary,independent increments ...

Think of return innovations () in discrete time: Rt+1 = t + tt+1.

Normal return innovation diffusionNon-normal return innovation jumps

Classic Levy specifications in finance:

Brownian motion (Black-Scholes, Merton)Compound Poisson process with normal jump size (Merton)

The return innovation distribution is either normal or mixture of normals.



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Outline

1 Levy processes


3 Examples

4 Evidence

5 Jump design


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Levy characteristics

Levy processes greatly expand our continuous-time choices of iid returninnovation distributions via the Levy triplet (,,(x)). ((x)Levydensity).

The Levy-Khintchine Theorem:

Xt(u) E

eiuXt

= et(u),(u) =

iu + 12 u

22 + R0 1eiux + iux1

|x

|


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Model stock returns with Levy processes

Let Xt be a Levy process, X(s) its cumulant exponent

The log return on a security can be modeled asln St/S0 = t + Xt tX(1)

where is the instantaneous drift (mean) of the stock such thatE[St] = S0e

t. The last term tX(1) is a convexity adjustment such thatXt

tX(1) forms an exponential martingale:

E

eXttX(1)

= 1.

Since both and X(1) are deterministic components, they can becombined together: ln St/S0 = mt + Xt, but it is more convenient to

separate them so that the mean instantaneous return is kept as aseparate free parameter.Under Q, = r q.Under this specification, we shall always set the first component of theLevy triplet to zero (0, , (x)), because it will be canceled out withthe convexity adjustment.

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Characteristic function of the security return

st ln St/S0 = t + Xt tX(1)

The characteristic function for the security return is

st(u) E eiu ln St/S0 = exp ( [iu + X(u) + iuX(1)] t)The characteristic exponent is

st(u) = iu + X(u) + iuX(1)

Under Q, = r q. The focus of the model specification is onXt (0, , (x)), unless r and/or q are modeled to be stochastic.

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Outline

1 Levy processes


3 Examples

4 Evidence

5 Jump design




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Tractable examples of Levy processes

1 Brownian motion (BSM) (t + Wt): normal shocks.

2

Compound Poisson jumps (Merton, 76): Large but rare events.

(x) = 1

2vJexp

(x J)

2

2vJ

.

3 Dampened power law (DPL):

(x) =

exp(+x) x

1

, x > 0, exp(|x|) |x|1, x < 0, , > 0, [1, 2)

Finite activity when < 0:R0

(x)dx < . Compound Poisson.Large and rare events.Infinite activity when

0: Both small and large jumps. Jump

frequency increases with declining jump size, and approaches infinity asx 0.Infinite variation when 1: many small jumps.

Market movements of all magnitudes, from small movements to market

crashes.

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Analytical characteristic exponents

Diffusion: (u) = iu + 12 u22.

Mertons compound Poisson jumps:

(u) =

1 eiuJ 12 u2vJ

.

Dampened power law: ( for = 0, 1)(u) = () (+ iu) + + ( + iu) iuC(h)When 2, smooth transition to diffusion (quadratic function of u).When = 0 (Variance-gamma by Madan et al):

(u) = ln (1 iu/+

) 1 + iu/ = ln(+ iu) ln + ln(

+ iu) ln

.

When = 1 (exponentially dampened Cauchy, Wu 2006):

(u) =

(+ iu) ln (+ iu) /+ +

+ iu

ln

+ iu/

iuC(h).



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The Black-Scholes model

The driver is a Brownian motion Xt = Wt.

We can write the return as

ln St/S0 = t + Wt 12

2t.

Note that (s) =1

2 s2

2

.The characteristic function of the return is:

(u) = exp

iut 1

2u22t iu1

22

= exp

iut 1

22

u2 + iu

t

.

Under Q, = r q.The characteristic exponent of the convexity adjusted Levy process(Xt X(1)t) is: X(u) + iuX(1) = 12 u22 + iu12 2 = 12 2(u2 + iu).

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Merton (1976)s jump-diffusion model

The driver of this model is a Levy process that has both a diffusion

component and a jump component.

The Levy triplet is (0, , (x)), with (x) = 12vJ

exp (xJ)22vJ

.

The first component of the triplet (the drift) is always normalized tozero.

The characteristic exponent of the Levy process isX(u) =

12 u

22 +

1 eiuJ 12 u2vJ

. The cumulant exponent is

X(s) =12 s

22 +

esJ+12 s

2vJ 1

.

We can write the return as ln St/S0 = t+ Xt

1

22 + eJ+ 12 vJ 1 t.

The characteristic function of the return is:

(u) = eiute12

2(u2+iu)te

1eiuJ 12 u2vJ

+iueJ+

12vJ1

t.

Under Q, = r q.

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Dampened power law (DPL)

The driver of this model is a pure jump Levy process, with its characteristicexponent

X(u) = () (+ iu) + + ( + iu) iuC(h). Thecumulant exponent isX(s) = ()

(+ s) + + ( + s)

+ sC(h)

We can write the return as, ln St/S0 = t + Xt X(1)t.

The characteristic function of the returnis:(u) = eiute

()

(+iu)

+ +

+iu

+iu()

(+1)

+ +

+1

t.

Under Q, = r q.The characteristic exponent of the convexity adjusted Levy process(Xt

X(1)t) is: X(u) + iuX(1).

References:

Carr, Geman, Madan, Yor, 2002, The Fine Structure of Asset Returns: An Empirical Investigation, Journal of

Business, 75(2), 305332.

Wu, 2006, Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns, Journal of

Business, 79(3), 14451474.

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Special cases of DPL

-stable law: No exponential dampening, = 0.Peter Carr, and Liuren Wu, Finite Moment Log Stable Process and Option Pricing, Journal of Finance, 2003, 58(2),

753777.

Without exponential dampening, return moments greater than are nolonger well defined.Characteristic function takes different form to account singularity.

Variance gamma (VG) model: = 0. Madan, Carr, Chang, 1998, The Variance Gamma Processand Option Pricing, European Finance Review, 2(1), 79105.

The characteristic exponent takes a different form as = 0 represents asingular point ((0) not well defined).

Double exponential model: = 1. Kou, 2002, A Jump-Diffusion Model for Option Pricing,Management Science, 48(8), 10861101.



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Other Levy examples

Other examples:

The normal inverse Gaussian (NIG) process of Barndorff-Nielsen (1998)The generalized hyperbolic process (Eberlein, Keller, Prause, 1998))The Meixner process (Schoutens, 2003))Jump to default model (Merton, 1976)

...Bottom line:

All tractable in terms of analytical characteristic exponents (u).

We can use FFT to generate the density function of the innovation (for

model estimation).

We can also use FFT to compute option values.


O li
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Outline

1 Levy processes


3 Examples

4 Evidence

5 Jump design



G l id L i i
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General evidence on Levy return innovations

Credit risk: (compound) Poisson process

The whole intensity-based credit modeling literature...Market risk: Infinite-activity jumps

Evidence from stock returns (CGMY (2002)): The estimates forDPL on most stock return series are greater than zero.Evidence from options: Models with infinite-activity return innovationsprice equity index options better (Carr & Wu (2003), Huang & Wu (2004))Li, Wells, & Yu (2006): Infinite-activity jumps cannot be approximatedby finite-activity jumps.

The role of diffusion (in the presence of infinite-variation jumps)

Not big, difficult to identify (CGMY (2002), Carr & Wu (2003a,b)).Generate correlations with diffusive activity rates (Huang & Wu (2004)).The diffusion (2) is identifiable in theory even in presence ofinfinite-variation jumps (At-Sahalia (2004), At-Sahalia&Jacod 2005).


I li d l ili il & k k
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Implied volatility smiles & skews on a stock

3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 20.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

AMD: 17Jan2006

Moneyness=ln(K/F)

ImpliedVolatility

Shortterm smile

Longterm skew

Maturities: 32 95 186 368 732


I li d l tilit k t k i d (SPX)
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Implied volatility skews on a stock index (SPX)

3 2.5 2 1.5 1 0.5 0 0.5 1 1.5 20.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

SPX: 17Jan2006

Moneyness=ln(K/F)

ImpliedVola

tility

More skews than smiles

Maturities: 32 60 151 242 333 704


A i li d l tilit il i
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Average implied volatility smiles on currencies

10 20 30 40 50 60 70 80 9011

11.5

12

12.5

13

13.5

14

Put delta

Averageimpliedvolatility

JPYUSD

10 20 30 40 50 60 70 80 908.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

Put delta

Averageimpliedvolatility

GBPUSD

Maturities: 1m (solid), 3m (dashed), 1y (dash-dotted)


Outline
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Outline

1 Levy processes


3

Examples

4 Evidence

5 Jump design



(I) The role of jumps at very short maturities
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(I) The role of jumps at very short maturities

Implied volatility smiles (skews)

non-normality (asymmetry) for the

risk-neutral return distribution.

IV(d) ATMV

1 +Skew.

6d +

Kurt.

24d2

, d =ln K/F

Two mechanisms to generate return non-normality:Use Levy jumps to generate non-normality for the innovationdistribution.Use stochastic volatility to generates non-normality through mixingover multiple periods.

Over very short maturities (1 period), only jumps contribute to returnnon-normalities.


Time decay of short term OTM options
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Time decay of short-term OTM options

Carr& Wu, What Type of Process Underlies Options? A Simple Robust Test, JF, 2003, 58(6), 25812610.

As option maturity

zero, OTM option value

zero.

The speed of decay is exponential O(ec/T) under pure diffusion, but linearO(T) in the presence of jumps.Term decay plot: ln(OTM/T) ln(T) at fixed K:

In the presence of jumps, the Black-Scholes implied volatility for OTM

options IV(, K) explodes as

0.


(II) The impacts of jumps at very long horizons


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(II) The impacts of jumps at very long horizons

Central limit theorem (CLT): Return distribution converge to normal withaggregation under certain conditions (finite return variance,...)

As option maturity increases, the smile should flatten.Evidence: The skew does not flatten, but steepens!

FMLS (Carr&Wu, 2003): Maximum negatively skewed -stable process.

Return variance is infinite. CLT does not apply.Down jumps only. Option has finite value.

But CLT seems to hold fine statistically:

0 5 10 15 201.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2

Time Aggregation, Days

Skewness

Skewness on S&P 500 Index Return

0 5 10 15 200

5

10

15

20

25

30

35

40

45

Time Aggregation, Days

Kurtosis

Kurtosis on S&P 500 Index Return


Reconcile P with Q via DPL jumps


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Reconcile P with Q via DPL jumps

Wu, Dampened Power Law: Reconciling the Tail Behavior of Financial Security Returns, Journal of Business, 2006, 79(3),

14451474.

Model return innovations under P by DPL:

(x) =

exp(+x) x1, x > 0, exp(|x|) |x|1, x < 0.

All return moments are finite with

> 0. CLT applies.

Market price of jump risk (): dQdP

t

= E(X)The return innovation process remains DPL under Q:

(x) = exp( (+ + ) x) x1, x > 0, exp( ( ) |x|) |x|

1

, x < 0.

To break CLT under Q, set = so that Q = 0.

Reconciling P with Q: Investors charge maximum allowed market price ondown jumps.


(III) Default risk & long-term implied vol skew
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(III) Default risk & long-term implied vol skew

When a company defaults, its stock value jumps to zero.

It generates a steep skew in long-term stock options.

Carr and Laurence (2006) approximation of the Merton (76) jump-to-defaultmodel:

IVt(d2, T) + N(d2)N(d2)

T t

The slope of the implied volatility smile atd

2 = 0 is

T

t.

Evidence: Stock option implied volatility skews are correlated with creditdefault swap (CDS) spreads written on the same company.

02 03 04 05 061

0

1

2

3

4

GM: Default risk and longterm implied volatility skew

Negative skewCDS spread

Carr & Wu, Stock Options and Credit Default Swaps: A Joint Framework for Valu ation and Estimation, JFEC, 2010.Liuren Wu (Baruch) Levy Processes Option Pricing 27 / 32

Three Levy jump components in stock returns


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Three Levy jump components in stock returns

I. Market risk (FMLS under Q, DPL under P)

The stock index skew is strongly negative at long maturities.

II. Idiosyncratic risk (DPL under both P and Q)

The smile on single name stocks is not as negatively skewed as that onindex at short maturities.

III. Default risk (Compound Poisson jumps).

Long-term skew moves together with CDS spreads.

Information and identification:

Identify market risk from stock index options.Identify the credit risk component from the CDS market.Identify the idiosyncratic risk from the single-name stock options.


Levy jump components in currency returns
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Levy jump components in currency returns

Model currency return as the difference of the log pricing kernels betweenthe two economies.

Pricing kernel assigns market prices to systematic risks.

Market risk dominates for exchange rates between two industrializedeconomies (e.g., dollar-euro).

Use a one-sided DPL for each economy (downward jump only).

Default risk shows up in FX for low-rating economies (say, dollar-peso).Peso drops by a large amount when the country (Mexico) defaults onits foreign debt.Peter Carr, and Liuren Wu, Theory and Evidence on the Dynamic Interactions Between Sovereign Credit

Default Swaps and Currency Options, Journal of Banking and Finance, 2007, 31(8), 23832403.

When pricing options on exchange rates, it is appropriate to distinguishbetween world risk versus country-specific risk.Bakshi, Carr, & Wu, Stochastic Risk Premiums, Stochastic Skewness in Currency Options, and Stochastic Discount

Factors in International Economies, JFE, 2008.


Outline
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Outline

1 Levy processes


3 Examples

4 Evidence

5 Jump design



Economic implications of using jumps
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Economic implications of using jumps

In the Black-Scholes world (one-factor diffusion):

The market is complete with a bond and a stock.The world is risk free after delta hedging.Utility-free option pricing. Options are redundant.

In a pure-diffusion world with stochastic volatility:

Market is complete with one (or a few) extra option(s).

The world is risk free after delta and vega hedging.

In a world with jumps of random sizes:

The market is inherently incomplete (with stocks alone).Need all options (+ model) to complete the market.Derman: Beware of economists with Greek symbols!Options market is informative/useful:

Cross sections (K,T) Q dynamics.Time series (t) P dynamics.The difference Q/P market prices of economic risks.


Bottom line
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Different types of jumps affect option pricing at both short and longmaturities.

Implied volatility smiles at very short maturities can only beaccommodated by a jump component.Implied volatility skews at very long maturities ask for a jump processthat generates infinite variance.Credit risk exposure may also help explain the long-term skew on single

name stock options.

The choice of jump types depends on the events:

Infinite-activity jumps frequent market order arrival.Finite-activity Poisson jumps rare events (credit).

The presence of jumps of random sizes creates value for the options markets...

Leve processes are largely static in the sense that they cannot generatetime variations in the return distribution and hence cannot accommodate

stochastic volatility, stochastic skewness, etc.Liuren Wu (Baruch) Levy Processes Option Pricing 32 / 32
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Using Levy Processes to Model Return Innovations